Theorem univ 4620

Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.)

Assertion

Ref	Expression
univ	⊢ ∪ V = V

Proof of Theorem univ

Step	Hyp	Ref	Expression
1		pwv 3932	. . 3 ⊢ 𝒫 V = V
2	1	unieqi 3943	. 2 ⊢ ∪ 𝒫 V = ∪ V
3		unipw 4355	. 2 ⊢ ∪ 𝒫 V = V
4	2, 3	eqtr3i 2261	1 ⊢ ∪ V = V

Syntax hints:   = wceq 1402  Vcvv 2821  𝒫 cpw 3688  ∪ cuni 3933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-uni 3934

This theorem is referenced by: (None)